The potential energy function for a diatomic molecule is `U(x) =(a)/(x^(12)) - (b)/(x^(6))`. In stable equilibrium, the distance between the particles is .

Options

A. `((2a)/(b))^(1//6)`
B. `(a/b)^(1//6)`
C. `((b)/(2a))^(1//6)`
D. `(b/a)^(1//6)`

Explanation

To find the distance between the particles in stable equilibrium for the given potential energy function \( U(x) = \frac{A}{x^{12}} - \frac{B}{x^{6}} \), we can follow these steps:

### Step 1: Understand the condition for stable equilibrium
In stable equilibrium, the force acting on the particles is zero. The force can be derived from the potential energy function as:
\[ F = -\frac{dU}{dx} \]
Setting this equal to zero gives us the condition for equilibrium:
\[ -\frac{dU}{dx} = 0 \]

### Step 2: Differentiate the potential energy function
We need to differentiate \( U(x) \):
\[ U(x) = \frac{A}{x^{12}} - \frac{B}{x^{6}} \]
Taking the derivative:
\[ \frac{dU}{dx} = -12 \frac{A}{x^{13}} + 6 \frac{B}{x^{7}} \]

### Step 3: Set the derivative equal to zero
Now, we set the derivative equal to zero to find the equilibrium position:
\[ -12 \frac{A}{x^{13}} + 6 \frac{B}{x^{7}} = 0 \]

### Step 4: Rearranging the equation
Rearranging the equation gives:
\[ 12 \frac{A}{x^{13}} = 6 \frac{B}{x^{7}} \]
Multiplying both sides by \( x^{13} \) to eliminate the denominator:
\[ 12A = 6B x^{6} \]

### Step 5: Solve for \( x^{6} \)
Now, we can solve for \( x^{6} \):
\[ x^{6} = \frac{12A}{6B} = \frac{2A}{B} \]

### Step 6: Find \( x \)
Taking the sixth root gives us the distance \( x \):
\[ x = \left(\frac{2A}{B}\right)^{\frac{1}{6}} \]

### Conclusion
Thus, the distance between the particles in stable equilibrium is:
\[ x = \left(\frac{2A}{B}\right)^{\frac{1}{6}} \]

---